Problema Solution
Need your help ASAP with a calculus problem
A roadway is to be constructed to connect an offshore oil well 4 miles from the shoreline
The oil company's field office is 10 miles along the shore from the point closest to the well
Cars can drive 70 mph on the road along the shorine but only 55 mph on the roadway that will be the bridge connecting the well and to the shore
Where should the bridge be constructed so that the driving time will be the shortest
Equations and differential?
Answer provided by our tutors
Since speed = distance/time follows time = distance/speed an we have:
The time needed to pass the distance of y miles driving with a rate of 55 mph is: y/55
The time needed to pass the distance of (10 - x) miles driving with a rate of 70 mph is: (10 - x)/70
Thus the total time is:
t = y/55 + (10 - x)/70
Now using the Pythagorean Theorem we know:
y^2 = x^2 + 4^2
y = √(x^2 + 4^2)
Plug y = √(x^2 + 4^2) into t = y/55 + (10 - x)/70:
t(x) = √(x^2 + 4^2)/55 + (10 - x)/70
We need to find the minimum thus we calculate the derivative and find the zeros:
t'(x) = √(x^2 + 4^2)/55 + (10 - x)/70 (click here to see the calculation of the derivative)
t'(x) = x/(55√(x^2 + 4^2)) - 1/70
t'(x) = 0
x/(55√(x^2 + 4^2)) - 1/70 = 0
........
click here to see the solution for the equation for x........
x = 6.10 mi
10 - x = 10 - 6.01 = 3.90 mi
The bridge should be constructed 3.90 miles from the filed office.